I know I'm overly paranoid but I won't do anything over hotel internet connections until I VPN into my home router first. Like others have said avoid business center computers as they may have keylogging software installed (not necessarily by the hotel, but it could be installed by other teenage guests.)

HTTPS connections do a quite good job protecting packets, but they don't protect you from tainted DNS servers (i.e., you enter "www.citibank.com" and the bad DNS resolves you to a phishing site which masquerades as the HTTPS version of the original). About 2 years ago I was in a major Vegas hotel and saw that Hotmail resolved to a private 10.10.x.x address instead of over the public internet.